Re: RE: To verify Cauchy-Riemann relations in complex variable graphically

*To*: mathgroup at smc.vnet.net*Subject*: [mg39228] Re: RE: To verify Cauchy-Riemann relations in complex variable graphically*From*: "Narasimham G.L." <google.news.invalid at web2news.net>*Date*: Tue, 4 Feb 2003 02:23:30 -0500 (EST)*References*: <b1l2k5$1@smc.vnet.net>*Reply-to*: "Narasimham G.L." <manoos2p+8ammthma18 at hotmail.com>*Sender*: owner-wri-mathgroup at wolfram.com

David Park: Thanks, I get error msg by run Show::gtype: CartesianMap is not a type of graphics. Is it version realated? mine is V 2.2 -- Narasimham > My friend Rip Pelletier pointed me to a better method to > illustrate the > Cauchy-Riemann relations for analytic functions. He > pointed me to Tristan > Needham's book 'Visual Complex Analysis'. In Chapter 5, Section I - > Cauchy-Riemann Revealed, the CR conditions are related to the complex > mapping. > > If a small patch of squares are mapped by an analytic > function, then they go > into another small patch in which all the squares have > been amplified and > rotated in exactly the same way. > > Fortunately, we have the ComplexMap package in Mathematica > and can easily > illustrate this for your functions. For example, for the > Sin function... > > Needs['Graphics`ComplexMap`'] > > With[ > {x = 2, > y = 2, > del = 0.01, > f = Sin}, > Show[GraphicsArray[{CartesianMap[ > Identity, {x - del, x + del, del/5}, {y - del, > y + del, del/5}, > Axes -> False, > DisplayFunction -> Identity], > > CartesianMap[ > f, {x - del, x + del, del/5}, {y - del, y + del, del/5}, > Axes -> False, > DisplayFunction -> Identity]}], > ImageSize -> 500]]; > > A square patch maps into a rotated square patch. Just > change f and/or the > mapping points for other cases. Use a pure function for z^2. > > For a case that is not analytic, and so the CR relations > do not hold, use > f = # + 2Abs[#] &. The squares go to parallelograms. > > David Park > djmp at earthlink.net > http://home.earthlink.net/~djmp/ > > From: Narasimham G.L. [mailto:google.news.invalid at web2news.net] To: mathgroup at smc.vnet.net > > Is it possible to have a semi transparent view of surfaces > so that one > may verify slopes by ParametricPlot3D for Cauchy-Riemann relations? > The following is program for 3 functions Z^2, Z^3, Sin[Z].It was > expected to check slopes at the line of intersection of Re > and Im parts. > > R1=x^2-y^2 ; I1= 2 x y ; > z2r=Plot3D[R1 , {x,-Pi,Pi},{y,-Pi,Pi} ]; > z2i=Plot3D[I1 , {x,-Pi,Pi},{y,-Pi,Pi} ]; > Show[z2r,z2i] ; 'Top view >> Re,Im Intxn'; > Plot[{x ArcTan[-Sqrt[2]+1],x ArcTan[Sqrt[2]+1]}, {x,-Pi,Pi} ]; > > R3=x^3 - 3 x y^2 ; I3= 3 x^2 y - y ^3 ; > z3r=Plot3D[R3 , {x,-Pi,Pi},{y,-Pi,Pi} ]; > z3i=Plot3D[I3 , {x,-Pi,Pi},{y,-Pi,Pi} ]; > Show[z3r,z3i] ; 'Top view >> Re,Im Intxn'; > Plot[{x,x (-Sqrt[3]+2) , x (-Sqrt[3]-2) }, {x,-Pi,Pi} ]; > > R2=Cosh[y] Sin[x] ; I2=Sinh[y] Cos[x] ; > scr=Plot3D[R2,{x,-Pi/2,Pi/2},{y,-Pi/2,Pi/2}]; > sci=Plot3D[I2,{x,-Pi/2,Pi/2},{y,-Pi/2,Pi/2}]; > Show[scr,sci]; 'Top view >> Re,Im Intxn'; > Plot[{ArcTanh[Tan[x]]},{x,-Pi/2,Pi/2 }]; -- Posted via http://web2news.com To contact in private, remove